<p>We exhibit the connection between the Wigner kernel and the Gabor matrix of a linear bounded operator <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2070_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="140" /> </InlineMediaObject> <EquationSource Format="TEX">\(T:\mathcal {S}(\mathbb {R}^d)\rightarrow \mathcal {S}'(\mathbb {R}^d)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>:</mo> <mi mathvariant="script">S</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <msup> <mrow> <mi mathvariant="script">S</mi> </mrow> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. The smoothing effect of the Gabor matrix is highlighted by basic examples. This connection allows a comparison between the classes of Fourier integral operators defined by means of the Gabor matrix in Cordero et al. (J Math Pures Appl 99(2):219-233, 2013) and the Wigner kernel in Cordero et al. (Nonlinear Differ Equ Appl 31:69, 2024. <a href="https://doi.org/10.3761007/s00030-024-00961-4">https://doi.org/10.3761007/s00030-024-00961-4</a>), showing the nice off-diagonal decay of the Gabor class with respect to the Wigner kernel and suggesting further investigations. Modulation spaces containing the Sjöstrand class are the symbol classes of this study.</p>